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Differential invariants of the 2D conformal Lie algebra action
(Master thesis; Mastergradsoppgave, 2008-02-15)
In this thises we consider the Lie algebra that corresponds to the Lie pseudogroup of all conformal transformations on the plane. This conformal Lie algebra is canonically represented as a Lie algebra of vector fields on R^2. We will find all possible representations of vector fields in R^3=J^0R^2 which projects to the canonical representation and find the algebra of scalar differential invariants ...
Extensions of groups and modules
(Master thesis; Mastergradsoppgave, 2010)
The main goal of this thesis is to build up detailed constructions and give complete proofs for the extension functors of modules and groups, which we define using cohomology functors. Further, we look at the relations that appear between these and short exact sequences of modules, respectively groups. We calculate also several concrete cohomology groups, and build extensions that are described by ...
Distributing a private key generator in Ad hoc Networks
(Master thesis; Mastergradsoppgave, 2009-05-15)
A Mobile Ad hoc Network (MANET) is a wireless network that does not rely on a fixed infrastructure. These characteristics make algorithms that route network traffic particularly vulnerable to attack. Mechanisms used to protect against such attacks often depend on cryptographic keys. Since the nodes in a MANET have limited resources, designing methods for cryptographic key management is ...
Kompleksiteten til noen kryptologisk viktige algoritmer
(Master thesis; Mastergradsoppgave, 1990-04-04)
Denne hovedfagsoppgaven har som mål å sammenligne teoretisk og praktisk kompleksitet til algoritmer som har stor kryptologisk betydning. Et av målene er å avsløre den skjulte konstanten bak O-notasjonen, slik at algoritmene kan sammenlignes på et reelt grunnlag. I tillegg er det sett på sammenheng mellom sikkerhet, størrelsen på tall, asymptotisk og praktisk kompleksitet. Spesielt algoritmer som ...
Numerical calculation of Casimir forces
(Master thesis; Mastergradsoppgave, 2012-06-19)
In this thesis a set of regularized boundary integral equation are introduced that can be used to calculate the Casimir force induced by a two dimensional scalar field. The boundary integral method is compared to the functional integral method and mode summation where possible. Comparisons are done for the case of two parallel plates, two concentric circles and two adjacent circles. The results ...
Local classification of 2-dimensional solvable Lie algebra actions on the plane.
(Master thesis; Mastergradsoppgave, 2010-05)
In the thesis the local classification of 2-dimensional solvable Lie algebra action on the plane is given. Normal forms of such actions are found. The classification applied to classifcation of 2nd order differential equations that are solvable in quadratures.
Hopf algebras and monoidal categories
(Master thesis; Mastergradsoppgave, 2007-06-14)
In this thesis we study the correspondence between categorical notions and bialgebra notions, and make a kind of dictionary and grammar book for translation between these notions. We will show how to obtain an antipode, and how to define braidings and quantizations. The construction is done in two ways. First we use the properties of a bialgebra to define a monoidal structure on (co)modules over ...
The use of elliptic curves in cryptography
(Master thesis; Mastergradsoppgave, 2007-06)
The use of elliptic curves in cryptography was suggested independently by Neal Koblitz and Victor Miller in 1985. Being a relatively new field, there is still a lot of ongoing research on the subject, but elliptic curve cryptography, or ECC for short, has already been implemented in real-life applications. Its strength was proved in 2003 when the U.S. National Security Agency adopted ECC for protecting ...
The Unidirectional Pulse Propagation Equation for Cylindrical Vector modes
(Master thesis; Mastergradsoppgave, 2015-07-27)
A new model for the unidirectional pulse propagation equations (UPPE) was developed by Per Jacobsen, this model is based on the assumption of cylindrical vector (CV) modes. The model will be strong for CV type electrical eld representations where only a few modes will be excited. In this thesis we will investigate the model further. The model will be implemented as a pseudo spectral method where ...
Arctic tipping points
(Master thesis; Mastergradsoppgave, 2015-05-15)
The Arctic is warming much faster than the entire planet, and this causes severe melting of sea ice. However, the climate of different regions of the Earth is interconnected, and changes in the amount of ice in the Arctic can dramatically affect the climate across the whole planet. Some scientists claim that a possible tipping point is the event of the ice-free Arctic Ocean in summer. Certain ...